I'm guessing an InfoPass appointment would help.
I have a similar (but different) problem. If you submit an N-400 6 months or more before your GC expires, you don't need to get a new one. We applied 8 months early. However, the USCIS took so long, our GCs expired.
It didn't seem to be a problem for use (we met the "more than 6 months rule"), but our (under 18) daughter's GC expired as well. As it expired, we applied for a new GC for her. We wanted to make sure she had a valid GC for her passport application and for the N-600 that we filled out for her.
Interestingly, her fingerprint appointment was the day before my wife took the oath. They put a sticker on her existing GC extending it's expiration until the end of this month. The next day she became a citizen.
She applied for a passport and submitted an N-600. Right about the time she got the passport, she got a shiny new GC.
At the moment, the USCIS still considers her a permanent resident, but the state department considers her a citizen. She currently has two valid green cards and a US passport. We figure we'll turn the new one in during the N-600 interview (when the USCIS gets around to that).
Oh well...
